Definition 6 For tensors \({X}\in \mathbb {R}^{m_1\times n_2 \dots \times n_d}\)  and \({Y}\in \mathbb {R}^{m^{\prime }_1\times n_2 \dots \times n_d},\)  their inner product in all modes except the first one [1]}, [2]} is defined as:
\(\mathbf {Z}= \langle {X}, {Y}\rangle _{-1} \in \mathbb {R}^{m_1 \times m^{\prime }_1}, ~~\text{where}~z_{i_1, i^{\prime }_1} = \sum _{j_d=1}^{n_d}\dots \sum _{j_2=1}^{n_2} x_{i_1,j_2, \dots , j_d}y_{i^{\prime }_1,j_2, \dots , j_d}, ~\forall ~i_1 \in [m_1], i^{\prime }_1 \in [m^{\prime }_1].\) 

